2.1.1 Introduction to Euclidean and Non-Euclidean
Geometry
PrintoutWithout the concepts, methods and results
found and developed by previous generations right down to Greek antiquity one
cannot understand either the aims or achievements of mathematics in the last 50
years.—Hermann Weyl (1885–1955)

Euclid's Fifth Postulate, the
parallel postulate, caused much dispute over many centuries. Many believed the
postulate should be a theorem and not an assumption. Eventually, non-Euclidean
geometries, based on postulates that were negations of Euclid's Fifth Postulate,
were formulated. Euclid the Thirteen Books of the Elements
translated by Thomas Heath is the best English translation used today of
Euclid's Elements. Here are links to two on-line editions of Euclid's
Elements: David E. Joyce's Java edition of
Euclid's
Elements (1997) or
Oliver
Byrne's edition of Euclid published in 1847. The problem with using
Euclid's
five axioms as a basis for a course in Euclidean geometry is that Euclid's
system has several flaws:
Euclid tried to define all terms and did not
recognize the need for undefined terms. Euclid made
other assumptions based on preconceptions that were not stated as postulates. And,
many proofs rely on diagrams and preconceptions about the diagrams.Two different, but equivalent, axiomatic
systems are used in the study of Euclidean geometry—synthetic geometry and
metric geometry. David Hilbert (1862–1943), in his book Gundlagen der
Geometrie (Foundations of Geometry), published in 1899 a list of
axioms for Euclidean geometry, which are axioms for a synthetic geometry.
Hilbert's axioms are in Appendix A of this chapter.
George Birkhoff (1884–1944) in a paper (A set of postulates for plane geometry
published in Annals of Mathematics in 1932) proposed a list of
axioms for Euclidean geometry, which were axioms for a metric geometry.
Birkhoff's axioms are in Appendix B of this chapter.
Read through and compare their axioms for Euclidean geometry. How are they
similar? How different?
The questions are: What axiom system should we
use in this course? How should we study them? Since this course is a survey
course for preparing teachers to teach high school geometry, we will use the
SMSG axiom set, which was developed by the School
Mathematics Study Group in 1961 as a suggestion for use with high school
geometry courses. Note how the SMSG axioms are a blend of Hilbert's and Birkhoff's axioms.
To show the similarities between Euclidean and non-Euclidean geometries, we will
postpone the introduction of a parallel postulate to the end of this chapter.
We will study what
is called neutral geometry, the properties
of which satisfy both Euclidean geometry and hyperbolic geometry. Then we will introduce
parallel postulates near the end of this chapter. Further, we will begin by
introducing several analytic models to illustrate and develop better
understanding of the axioms and concepts. Also, we will restrict our study to
plane geometry and forgo the axioms for space. We will use the plane
geometry axioms from the SMSG axiom set.
This chapter assumes the reader has had at least a
high school Euclidean geometry course that included proofs.
Use prepared software from
Appendix B of the Course Title Page
Prepared Geometer's Sketchpad Sketches,
and NonEuclidjava program for constructions in hyperbolic geometry
http://cs.unm.edu/~joel/NonEuclid/NonEuclid.html.

Before we begin a formal study of neutral geometry
leading to Euclidean and hyperbolic geometry, we emphasize the importance of the
axioms and not making additional assumptions based on diagrams.